# An Iterative Vertex Enumeration Method for Objective Space Based Vector Optimization Algorithms

## TL;DR

This paper introduces an iterative vertex enumeration method tailored for objective space-based vector optimization algorithms, improving efficiency especially in higher dimensions by modifying the double description method for unbounded polyhedrons.

## Contribution

The paper presents a novel iterative vertex enumeration procedure that enhances existing algorithms for vector optimization by efficiently handling unbounded polyhedrons in the objective space.

## Key findings

- Modified DD method outperforms existing methods in higher dimensions.
- The new procedure reduces computation time for large multiobjective problems.
- Performance improvements are validated through tests on randomly generated problems.

## Abstract

An application area of vertex enumeration problem (VEP) is the usage within objective space based linear/convex {vector} optimization algorithms whose aim is to generate (an approximation of) the Pareto frontier. In such algorithms, VEP, which is defined in the objective space, is solved in each iteration and it has a special structure. Namely, the recession cone of the polyhedron to be generated is the {ordering} cone. We {consider and give a detailed description of} a vertex enumeration procedure, which iterates by calling a modified `double description (DD) method' that works for such unbounded polyhedrons. We employ this procedure as a function of an existing objective space based {vector} optimization algorithm (Algorithm 1); and test the performance of it for randomly generated linear multiobjective optimization problems. We compare the efficiency of this procedure with another existing DD method as well as with the current vertex enumeration subroutine of Algorithm 1. We observe that the modified procedure excels the others especially as the dimension of the vertex enumeration problem (the number of objectives of the corresponding multiobjective problem) increases.